Spectral Analysis of Stochastic Neural Oscillators

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Many aspects of brain function involve the coordinated activity of millions of nerve cells. Individually, these cells can behave like tiny clocks, emitting a steady stream of pulses that communicate with each other. The synchronization of these oscillators ("clocks") plays a role in both healthy and diseased brain states. Deepening the understanding of how groups of brain cells synchronize ("tick" together) or desynchronize ("tick" separately) will deepen the understanding of the brain systems underlying motor control, epilepsy, breathing, information processing, and cognition. This project addresses a conceptual gap in the theoretical understanding of oscillating nerve cells. Today, most theories about how nerve cells synchronize rely on the assumption that each cell behaves with nearly impeccable precision. However, real nerve cells have stochastic variability, and their behavior is partly irregular and unpredictable. In constructing mathematical models of nerve cells that can account for variability, mathematically challenging problems arise. Solving these mathematical problems can improve the ability to quantitatively describe the clocklike behavior of individual nerve cells. This project will contribute to the BRAIN Initiative by providing deeper insight into many nervous system disorders.This project addresses a mathematical problem at the foundations of computational neuroscience and neurophysiology: the reduction of oscillatory systems to a phase oscillator description, in the presence of noise. The asymptotic phase of a deterministic dynamical system possessing a stable limit cycle was introduced in mathematical biology in the 1970s. Today, the biological significance of phase oscillator models is difficult to overstate. The fundamental notions of phase and phase resetting depend on classical results of invariant manifold theory for systems of deterministic differential equations. But noise is ubiquitous in biological dynamics. In recasting models of biological oscillators to take into account random fluctuations, the classical "asymptotic phase" is no longer well defined. This project develops a new definition for the asymptotic phase of a noisy oscillator, defined in terms of the complex eigenfunctions of the forward Kolmogorov operator, describing the evolution of the density on the state space. This "stochastic asymptotic phase" coincides with the classical phase in the case of vanishing noise intensities. However this new definition does not rely on the classical phase of a related deterministic system. Unlike the classical phase, it is equally well defined for systems that require noise for sustained oscillation.